Foundations of classical electrodynamics

458 Pages·2003·1.867 MB·English

by  F. W. Hehl, Yuri N. Obukhov

Foundations of Classical
1;2
Electrodynamics
Friedrich W. Hehl & Yuri N. Obukhov
01 June 2001
1
Keywords: electrodynamics,electromagnetism, axiomatics, ex-
terior calculus, classical (cid:12)eld theory, (coupling to) gravitation, com-
puteralgebra.{ThebookiswritteninAmericanEnglish(orwhatthe
authors conceive as such). Allfootnotes inthe bookwillbe collected
at the end of each Part before the references. The image created by
Glatzmaier (Fig. B.3.4) is in color, possibly also Fig. B.5.1 on the
aspects of the electromagnetic (cid:12)eld. The present format of the book
should be changed such as to allow for a 1-line display of longer
mathematical formulas.
2
The book is written in Latex. Our master(cid:12)le is birk.tex.
The di(cid:11)erent part of the books correspond to the (cid:12)les partI.tex,
partA.tex, partB.tex, partC.tex, partD.tex, partE.tex,
partO.tex. As of today, only a truncated version is available of the
outlook chapter on the (cid:12)le partO.trunc.tex. The completed ver-
sion will be handed in later as (cid:12)le partO.tex. { The (cid:12)gures are on
separate (cid:12)les. Their names are presently given on the (cid:12)rst page of
the text of each part of an outprint. For some of the (cid:12)gures we have
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attached to the draft versionofthe book.Followupofcommandson
a Unix system: cd birk, latex birk (2 times), makeindex birk, latex
birk, dvips -f birk > birk.ps, gv birk.ps&.
Contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . 5
Introduction . . . . . . . . . . . . . . . . . . . . . . . 8
Five plus one axioms . . . . . . . . . . . . . . . . 8
Topological approach . . . . . . . . . . . . . . . . 10
Electromagnetic spacetime relation as (cid:12)fth axiom 11
Electrodynamics in matter and the sixth axiom . 13
List of axioms . . . . . . . . . . . . . . . . . . . . 13
Areminder:Electrodynamicsin3-dimensionalEu-
clidean vector calculus . . . . . . . . . . . 13
On the literature . . . . . . . . . . . . . . . . . . 16
References 19
A Mathematics: Some exterior calculus 26
Why exterior di(cid:11)erential forms? 27
A.1 Algebra 33
A.1.1 A real vector space and its dual . . . . . . . . . 33
p
A.1.2 Tensors of type q . . . . . . . . . . . . . . . . . 37
(cid:2) (cid:3)
vi Contents
A.1.3 (cid:10)A generalization of tensors: geometric quantities 38
A.1.4 Almost complex structure . . . . . . . . . . . . . 40
A.1.5 Exterior p-forms . . . . . . . . . . . . . . . . . . 41
A.1.6 Exterior multiplication . . . . . . . . . . . . . . 43
A.1.7 Interior multiplication of a vector with a form. . 46
A.1.8 (cid:10)Volume elements on a vector space, densities,
orientation . . . . . . . . . . . . . . . . . . . . . . 47
A.1.9 (cid:10)Levi-CivitasymbolsandgeneralizedKronecker
deltas . . . . . . . . . . . . . . . . . . . . . . . . 51
6
A.1.10The space M of two-forms in four dimensions . 55
6
A.1.11Almost complex structure on M . . . . . . . . . 60
A.1.12 Computer algebra . . . . . . . . . . . . . . . . . 63
A.2 Exterior calculus 77
A.2.1 (cid:10)Di(cid:11)erentiable manifolds . . . . . . . . . . . . . 78
A.2.2 Vector (cid:12)elds . . . . . . . . . . . . . . . . . . . . 82
A.2.3 One-form (cid:12)elds, di(cid:11)erential p-forms . . . . . . . 83
A.2.4 Images of vectors and one-forms . . . . . . . . . 84
A.2.5 (cid:10)Volume forms and orientability . . . . . . . . . 87
A.2.6 (cid:10)Twisted forms . . . . . . . . . . . . . . . . . . 88
A.2.7 Exterior derivative . . . . . . . . . . . . . . . . . 90
A.2.8 Frame and coframe . . . . . . . . . . . . . . . . 93
A.2.9 (cid:10)Maps of manifolds: push-forward and pull-back 95
A.2.10(cid:10)Lie derivative . . . . . . . . . . . . . . . . . . . 97
A.2.11Excalc, a Reduce package . . . . . . . . . . . . . 104
A.2.12(cid:10)Closed and exact forms, de Rham cohomology
groups . . . . . . . . . . . . . . . . . . . . . . . . 108
A.3 Integration on a manifold 113
A.3.1 Integrationof0-formsandorientabilityofaman-
ifold . . . . . . . . . . . . . . . . . . . . . . . . . 113
A.3.2 Integration of n-forms . . . . . . . . . . . . . . . 114
A.3.3 Integration of p-forms with 0 < p < n . . . . . . 116
A.3.4 Stokes’s theorem . . . . . . . . . . . . . . . . . . 121
A.3.5 (cid:10)De Rham’s theorems . . . . . . . . . . . . . . . 124
References 131
Contents vii
B Axioms of classical electrodynamics 136
B.1 Electric charge conservation 139
B.1.1 Countingcharges.Absoluteandphysicaldimen-
sion . . . . . . . . . . . . . . . . . . . . . . . . . 139
B.1.2 Spacetime and the (cid:12)rst axiom . . . . . . . . . . 145
B.1.3 Electromagnetic excitation . . . . . . . . . . . . 147
B.1.4 Time-spacedecompositionoftheinhomogeneous
Maxwell equation . . . . . . . . . . . . . . . . . . 148
B.2 Lorentz force density 153
B.2.1 Electromagnetic (cid:12)eld strength . . . . . . . . . . 153
B.2.2 Second axiom relating mechanics
and electrodynamics . . . . . . . . . . . . . . . . 155
B.2.3 The (cid:12)rst three invariants of the electromagnetic
(cid:12)eld . . . . . . . . . . . . . . . . . . . . . . . . . 157
B.3 Magnetic (cid:13)ux conservation 161
B.3.1 Third axiom . . . . . . . . . . . . . . . . . . . . 161
B.3.2 Electromagnetic potential . . . . . . . . . . . . . 166
B.3.3 Abelian Chern-Simons and Kiehn 3-forms . . . . 167
B.3.4 Measuring the excitation . . . . . . . . . . . . . 169
B.4 Basic classical electrodynamics summarized,
example 177
B.4.1 Integral version and Maxwell’s equations . . . . 177
B.4.2 (cid:10)Jumpconditionsforelectromagneticexcitation
and (cid:12)eld strength . . . . . . . . . . . . . . . . . . 183
B.4.3 Arbitrarylocalnon-inertialframe:Maxwell’sequa-
tions in components . . . . . . . . . . . . . . . . . 184
B.4.4 (cid:10)Electrodynamicsin(cid:13)atland:2-dimensionalelec-
tron gas and quantum Hall e(cid:11)ect . . . . . . . . . 186
B.5Energy-momentum current and action 199
B.5.1 Fourth axiom: localization of energy-momentum 199
B.5.2Propertiesofenergy-momentum,electric-magnetic
reciprocity . . . . . . . . . . . . . . . . . . . . . . 202
B.5.3Time-space decomposition of energy-momentum . 212
viii Contents
B.5.4 Action . . . . . . . . . . . . . . . . . . . . . . . 214
B.5.5 (cid:10)Coupling of the energy-momentum current to
the coframe . . . . . . . . . . . . . . . . . . . . . 218
B.5.6 Maxwell’s equations and the energy-momentum
current in Excalc . . . . . . . . . . . . . . . . . . 222
References 227
C More mathematics 232
C.1 Linear connection 233
C.1.1 Covariant di(cid:11)erentiation of tensor (cid:12)elds . . . . . 234
C.1.2 Linear connection 1-forms . . . . . . . . . . . . . 236
C.1.3 (cid:10)Covariantdi(cid:11)erentiationofageneralgeometric
quantity . . . . . . . . . . . . . . . . . . . . . . . 239
C.1.4 Parallel transport . . . . . . . . . . . . . . . . . 240
C.1.5 (cid:10)Torsion and curvature . . . . . . . . . . . . . . 241
C.1.6 (cid:10)Cartan’sgeometricinterpretationoftorsionand
curvature . . . . . . . . . . . . . . . . . . . . . . 246
C.1.7 (cid:10)Covariant exterior derivative . . . . . . . . . . 248
C.1.8 Thep-formso(a),conn1(a,b),torsion2(a),curv2(a,b)250
C.2 Metric 253
C.2.1 Metric vector spaces . . . . . . . . . . . . . . . . 254
C.2.2 (cid:10)Orthonormal, half-null, and null frames, the
coframe statement . . . . . . . . . . . . . . . . . 256
C.2.3 Metric volume 4-form . . . . . . . . . . . . . . . 260
C.2.4 Duality operator for 2-forms as a symmetric al-
6
most complex structure on M . . . . . . . . . . . 262
C.2.5 Fromthedualityoperatortoatripletofcomplex
2-forms . . . . . . . . . . . . . . . . . . . . . . . . 264
C.2.6 From the triplet of complex 2-forms to a duality
operator . . . . . . . . . . . . . . . . . . . . . . . 266
C.2.7 From a triplet of complex 2-forms to the metric:
Scho(cid:127)nberg-Urbantke formulas . . . . . . . . . . . 269
C.2.8 Hodge star and Excalc’s # . . . . . . . . . . . . 271
C.2.9 Manifold with a metric, Levi-Civita connection . 275
Contents ix
C.2.10Codi(cid:11)erential and wave operator, also in Excalc 277
C.2.11(cid:10)Nonmetricity . . . . . . . . . . . . . . . . . . . 279
C.2.12(cid:10)Post-Riemannian pieces of the connection . . . 281
C.2.13 Excalc again . . . . . . . . . . . . . . . . . . . . 285
References 291
D The Maxwell-Lorentz spacetime
relation 293
D.1 Linearity between H and F and quartic wave
surface 295
D.1.1 Linearity . . . . . . . . . . . . . . . . . . . . . . 295
D.1.2 Extracting the Abelian axion . . . . . . . . . . . 298
D.1.3 Fresnel equation . . . . . . . . . . . . . . . . . . 300
D.1.4 Analysis of the Fresnel equation . . . . . . . . . 305
D.2 Electric-magnetic reciprocity switched on 311
D.2.1 Reciprocity implies closure . . . . . . . . . . . . 311
D.2.2 Almost complex structure . . . . . . . . . . . . . 313
D.2.3 Algebraic solution of the closure relation . . . . 314
D.3 Symmetry switched on additionally 317
D.3.1 Lagrangian and symmetry . . . . . . . . . . . . 317
D.3.2 Duality operator and metric . . . . . . . . . . . 319
D.3.3 (cid:10)Algebraicsolutionoftheclosureandsymmetry
relations . . . . . . . . . . . . . . . . . . . . . . . 320
D.3.4 From a quartic wave surface to the lightcone . . 326
D.4 Extracting the conformally invariant part of
the metric by an alternative method 333
D.4.1 (cid:10)Triplet of self-dual 2-forms and metric . . . . . 334
D.4.2 Maxwell-LorentzspacetimerelationandMinkowski
spacetime . . . . . . . . . . . . . . . . . . . . . . 337
D.4.3 Hodge star operator and isotropy . . . . . . . . . 338
D.4.4 (cid:10)Covariance properties . . . . . . . . . . . . . . 340
x Contents
D.5 Fifth axiom 345
References 347
E Electrodynamics in vacuum and in mat-
ter 352
E.1 Standard Maxwell{Lorentz theory in vacuum 355
E.1.1 Maxwell-Lorentzequations,impedanceofthevac-
uum . . . . . . . . . . . . . . . . . . . . . . . . . 355
E.1.2 Action . . . . . . . . . . . . . . . . . . . . . . . 357
E.1.3 Foliationof a spacetime with a metric. E(cid:11)ective
permeabilities . . . . . . . . . . . . . . . . . . . . 358
E.1.4 Symmetry of the energy-momentum current . . . 360
E.2 Electromagnetic spacetime relations beyond lo-
cality and linearity 363
E.2.1 Keeping the (cid:12)rst four axioms (cid:12)xed . . . . . . . . 363
E.2.2 (cid:10)Mashhoon . . . . . . . . . . . . . . . . . . . . . 364
E.2.3 Heisenberg-Euler . . . . . . . . . . . . . . . . . . 365
E.2.4 (cid:10)Born-Infeld . . . . . . . . . . . . . . . . . . . . 366
E.2.5 (cid:10)Pleban(cid:19)ski . . . . . . . . . . . . . . . . . . . . . 367
E.3 Electrodynamics in matter, constitutive law 369
E.3.1 Splitting of the current: Sixth axiom . . . . . . . 369
E.3.2 Maxwell’s equations in matter . . . . . . . . . . 371
E.3.3 Linear constitutive law . . . . . . . . . . . . . . 372
E.3.4 Energy-momentum currents in matter . . . . . . 373
E.3.5 (cid:10)Experiment of Walker & Walker . . . . . . . . 379
E.4 Electrodynamics of moving continua 383
E.4.1 Laboratory and material foliation . . . . . . . . 383
E.4.2 Electromagnetic(cid:12)eldinlaboratoryandmaterial
frames . . . . . . . . . . . . . . . . . . . . . . . . 387
E.4.3 Optical metric from the constitutive law . . . . . 391
E.4.4 Electromagnetic (cid:12)eld generated in moving con-
tinua . . . . . . . . . . . . . . . . . . . . . . . . . 392
Contents xi
E.4.5 The experiments of R(cid:127)ontgenand Wilson&Wilson396
E.4.6 Non-inertial \rotating coordinates" . . . . . . . . 401
E.4.7 Rotating observer . . . . . . . . . . . . . . . . . 403
E.4.8 Accelerating observer . . . . . . . . . . . . . . . 405
E.4.9 The proper reference frame of the noninertial
observer (\noninertial frame") . . . . . . . . . . . 408
E.4.10 Universality of the Maxwell-Lorentz spacetime
relation . . . . . . . . . . . . . . . . . . . . . . . 410
References 413
F Preliminarysketchversion of Validity of clas-
sical electrodynamics, interaction with grav-
ity, outlook 416
F.1 Classical physics (preliminary) 419
F.1.1 Gravitational (cid:12)eld . . . . . . . . . . . . . . . . . 419
F.1.2 Classical (1st quantized) Dirac (cid:12)eld . . . . . . . 430
F.1.3 Topology and electrodynamics . . . . . . . . . . 432
F.1.4 Remark on possible violations of Poincar(cid:19)e in-
variance . . . . . . . . . . . . . . . . . . . . . . . 435
F.2 Quantum physics (preliminary) 437
F.2.1 QED . . . . . . . . . . . . . . . . . . . . . . . . 437
F.2.2 Electro-weak uni(cid:12)cation . . . . . . . . . . . . . . 438
F.2.3 Quantum Chern-Simons and the QHE . . . . . . 440
References 441
Index 444
Contents 1
(cid:12)lebirk/partI.tex,with(cid:12)gures[I01cover.eps,I02cover.eps],2001-
06-01
Description of the book
Electric and magnetic phenomena are omnipresent in modern
life. Their non-quantum aspects are successfully described by
classicalelectrodynamics(Maxwell’stheory).Inthisbook,which
is an outgrowth of a physics graduate course, the fundamental
structure of classicalelectrodynamicsis presented inthe form of
six axioms: (1) electric charge conservation, (2) existence of the
Lorentz force, (3) magnetic (cid:13)ux conservation, (4) localization
of electromagnetic energy-momentum, (5) existence of an elec-
tromagnetic spacetime relation, and (6) splitting of the electric
current in material and external pieces.
The (cid:12)rst four axioms are well-established. For their formu-
lation an arbitrary 4-dimensional di(cid:11)erentiable manifold is re-
quired which allows for a foliation into 3-dimensional hypersur-
faces. The (cid:12)fth axiom characterizes the environment in which
the electromagnetic (cid:12)eld propagates, namely spacetime with or
without gravitation. The relativistic description of such general
environments remains a research topic of considerable interest.
In particular, it is only in this (cid:12)fth axiom that the metric ten-
sor of spacetime makes its appearance, thus coupling electro-
magnetism and gravitation. The operational interpretation of
the physical notions introduced is stressed throughout. In par-
ticular, the electrodynamics of moving matter is developed ab
initio.
The tool for formulating the theory is the calculus of exterior
di(cid:11)erential forms which is explained in suÆcient detail, includ-
ing the corresponding computer algebra programs.
This book presents a fresh and original exposition of the
foundations of classical electrodynamics in the tradition of the
so-called metric-free approach. The reader will win a new out-
lookonthe interrelationshipand the inner workingof Maxwell’s
equations and their raison d’^etre.
2 Contents
Book Announcement (old version of 1999):
Electric and magnetic phenomena play an important role in
the natural sciences. They are described by means of classical
electrodynamics, as formulated by Maxwell in 1864. As long as
the electromagnetic (cid:12)eld is not quantized, Maxwell’s equations
provide a correct description of electromagnetism.
In this work the foundations of classical electrodynamics are
displayed in a consistent axiomatic way. The presentation is
based on the simple but far-reaching axioms of electric charge
conservation, the existence of the Lorentz force, magnetic (cid:13)ux
conservation, and the localization of energy-momentum. While
the (cid:12)rst four axioms above are well-established, they are insuf-
(cid:12)cient to complete the theory. The missing ingredient is a char-
acterization of the environment in which the electromagnetic
(cid:12)eld propagates by means of constitutive relations. This envi-
ronment is spacetime with or without a material medium and
with or without gravitation (curvature, perhaps torsion). The
relativistic description of such general environments remains a
research topic of considerable interest. In particular, it is only
in this last axiom that the spacetime metric tensor makes its
appearance, thus coupling electromagnetism and gravitation.
The appropriate tool for this theory is the calculus of exte-
rior di(cid:11)erential forms, which is introduced before the axioms of
electrodynamics are formulated. The operational interpretation
of the physical notions introduced is stressed throughout.
The book may be used for a course or seminar in theoreti-
cal electrodynamics by advanced undergraduates and graduate
students inmathematics, physics, and electrical engeneering.Its
approach to the fundamentals of classical electrodynamics will
also be of interest to researchers and instructors as well in the
above-mentioned (cid:12)elds.
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